On the Asymptotic Expansion of Hadamard Products

We consider functions f and g which are holomoxphic on closed sectors in C where they admit an asymptotic representation at ∞ in the form of power series in z^-1. We give a simple geometrical condition under which the Hadamard product f*g of f and g porsesses again an asymp totic expansion at ∞. It turns out that the asymptotic expansion of f*g is essentially the formal Hsdamd product of the asymptotic expansions of f and g. Our result yields a slight generaliestion of a well known theorem of W. B. Ford.     

A FUNCTIONAL CENTRAL LIMIT THEOREMFOR NEGATIVELY ASSOCIATED SEQUENCE

Let(x1,j≥1)be a sequence of negatively associated random variables with ex1=o,ex^21＜∞.in this paper a functional central limit theorem for negatively associated random variables under some conditions withbout stationarity is proved which is the same as the results for positively associated random variables.     

A Note on the Browder＇s and Weyl＇s Theorem

Let T be a Banach space operator, E（T） be the set of all isolated eigenvalues of T and π（T） be the set of all poles of T. In this work, we show that Browder＇s theorem for T is equivalent to the localized single-valued extension property at all complex numbers λ in the complement of the Weyl spectrum of T, and we give some characterization of Weyl＇s theorem for operator satisfying E（T） = π（T）. An application is also given.     

Dirac's Condition for Completely Independent Spanning Trees

Two spanning trees T₁ and T₂ of a graph G are completely independent if, for any two vertices u and v, the paths from u to v in T₁ and T₂ are internally disjoint. In this article, we show two sufficient conditions for the existence of completely independent spanning trees. First, we show that a graph of n vertices has two completely independent spanning trees if the minimum degree of the graph is at least . Then, we prove that the square of a 2‐connected graph has two completely independent spanning trees. These conditions are known to be sufficient conditions for Hamiltonian graphs.     

On existence of solutions of multivalued stochastic differential equations with discontinuous coefficients

The subject of the paper is to find existence conditions of weak solutions to multivalued stochastic differential equations with discontinuous coefficients. First we prove that a non-exploding solution exists when the drift coefficient b satisfies linear growth and the diffusion coefficient σ is uniformly elliptic. On this basis, we continue to obtain a solution (up to the explosion time) in the weak sense under certain local integrability, improving the result of Rozkosz and S?omiński.     

Two-sided bounds and leading term for rates of convergence in the multivariate central limit theorem

For a sequence of independent and identically distributed random vectors, upper and lower bounds are obtained for the discrepancy between the probability measure P_n, induced by their normalized sum, and the Normal measure Φ. The upper and lower bounds are of the same order of magnitude. These results may be derived by a “leading term” approach, in which a signed measure Q_n is introduced as a first order approximation to P_n − Φ. The purpose of this paper is to investigate properties of the leading term.     

The Weak Convergence for Functions of Negatively Associated Random Variables

Let {X_n, n1} be a sequence of stationary negatively associated random variables, S_j(l)=∑^l_i=1 X_j+i, S_n=∑ⁿ_i=1 X_i. Suppose that f(x) is a real function. Under some suitable conditions, the central limit theorem and the weak convergence for sums are investigated. Applications to limiting distributions of estimators of Var S_n are also discussed.     

On Positive Random Objects

The idea of defining the expectation of a random variable as its integral with respect to a probability measure is extended to certain lattice-valued random objects and basic results of integration theory are generalized. Conditional expectation is defined and its properties are developed. Lattice valued martingales are also studied and convergence of sub- and supermartingales and the Optional Sampling Theorem are proved. A martingale proof of the Strong Law of Large Numbers is given. An extension of the lattice is also studied. Studies of some applications, such as on random compact convex sets in R ⁿ and on random positive upper semicontinuous functions, are carried out, where the generalized integral is compared with the classical definition. The results are also extended to the case where the probability measure is replaced by a -finite measure.     

Rate of convergence in the central limit theorem for empirical processes

Let _n and be an empirical process and a generalized Brownian bridge, respectively, indexed by a class of real measurable functions. From the central limit theorem for empirical processes it follows that for allr0. In this paper, assuming the class to be countably determined, under certain conditions we obtain an estimate for some constantC. Vapnik-ervonenkis class and the indicators of lower left orthants provide examples of classes considered here.     

Normal approximation and fourth moment theorems for monochromatic triangles

Given a graph sequence denote by T₃(G_n) the number of monochromatic triangles in a uniformly random coloring of the vertices of G_n with colors. In this paper we prove a central limit theorem (CLT) for T₃(G_n) with explicit error rates, using a quantitative version of the martingale CLT. We then relate this error term to the well-known fourth-moment phenomenon, which, interestingly, holds only when the number of colors satisfies . We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any , which, together with the above result, implies that the fourth-moment condition characterizes the limiting normal distribution of T₃(G_n), whenever . Finally, to illustrate the promise of our approach, we include an alternative proof of the CLT for the number of monochromatic edges, which provides quantitative rates for the results obtained in [7].     

Analogs of the Kolmogorov-Seliverstov-Plessner theorem for nonorthogonal function systems

For arbitrary function systems, the growth of partial sums is estimated depending on the growth of the corresponding Lebesgue functions. We prove analogs of the Kolmogorov-Seliverstov-Plessner convergence theorem for trigonometric series and the Kaczmarz convergence theorem for orthogonal series for arbitrary (nonorthogonal) function systems as well as for orthogonal-like and generalized orthogonal-like systems. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 87–101, January, 2000.     

On the convergence of the LJ search method

The convergence of the Luus-Jaakola search method for unconstrained optimization problems is established.Notation E _n Euclideann-space - f Gradient off(x) - ² f Hessian matrix - (·) ^T Transpose of (·) - I Index set {1, 2, ...,n} - [x _i1 ^*(j) ] Point around which search is made in the (j + 1)th iteration, i.e., [x _1l ^*(j) ,x _2l ^*(j) ,...,x _n1 ^*(j) ] - r _i ⁽ⁱ⁾ Range ofx _il ^*(i) in the (j + 1)th iteration - l ₁ min_i {r _i ⁽⁰⁾ } - l ₂ min_i {r _i ⁽⁰⁾ } - A _j Region of search in thejth iteration, i.e., {x E _n:x _il ^*(j-1) –0.5r _i ^(j-1) x _ix _il ^*(j-1) +0.5r _i ^(j-1) ,i I} - S _j Closed sphere with center origin and radius _j - Reduction factor in each iteration - 1– - (·) Gamma function Many discussions with Dr. S. N. Iyer, Professor of Electrical Engineering, College of Engineering, Trivandrum, India, are gratefully acknowledged. The author has great pleasure to thank Dr. K. Surendran, Professor, Department of Electrical Engineering, P.S.G. College of Technology, Coimbatore, India, for suggesting this work.     

On Bohr's theorem for multiple Fourier series

The problem of Pringsheim uniform convergence of multiple Fourier series in the trigonometric system is considered. A multidimensional analog of Bohr's theorem on the uniform convergence of the Fourier series of a continuous function after a homeomorphic chance of variable is proved.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 913–924, December, 1998.     

Distribution of a certain partition function modulo powers of primes

In this paper, we study a certain partition function a(n) defined by Σ _n≥0 a(n)q ⁿ := Π _n=1(1 − q ⁿ )⁻¹(1 − q ²ⁿ )⁻¹. We prove that given a positive integer j ≥ 1 and a prime m ≥ 5, there are infinitely many congruences of the type a(An + B) ≡ 0 (mod m ^j ). This work is inspired by Ono’s ground breaking result in the study of the distribution of the partition function p(n).     

Korovkin type approximation theorem via Ka−convergence on weighted spaces

In this paper, we study the Korovkin type approximation theorem for K_a‐ convergence, which is an interesting convergence method on weighted spaces. We also study the rate of K_a?convergence by using the weighted modulus of continuity and afterwards, we present a nontrivial application.     

A Remark on Chen’s Theorem (II)

Let p denote a prime and P2 denote an almost prime with at most two prime factors. The author proves that for sufficiently large x,∑p≤x p＋2=P2 1〉1.13Cx/log^2x,where the constant 1.13 constitutes an improvement of the previous result 1.104 due to J. Wu.     

Skorohod representation on a given probability space

Let $(\Omega,\mathcal{A},P)Let be a probability space, S a metric space, μ a probability measure on the Borel σ-field of S, and an arbitrary map, n = 1,2,.... If μ is tight and X _n converges in distribution to μ (in Hoffmann–J?rgensen’s sense), then X∼μ for some S-valued random variable X on . If, in addition, the X _n are measurable and tight, there are S-valued random variables and X, defined on , such that , X∼μ, and a.s. for some subsequence (n _k ). Further, a.s. (without need of taking subsequences) if μ{x} = 0 for all x, or if P(X _n = x) = 0 for some n and all x. When P is perfect, the tightness assumption can be weakened into separability up to extending P to for some H⊂Ω with P ^*(H) = 1. As a consequence, in applying Skorohod representation theorem with separable probability measures, the Skorohod space can be taken , for some H⊂ (0,1) with outer Lebesgue measure 1, where is the Borel σ-field on (0,1) and m _H the only extension of Lebesgue measure such that m _H (H) = 1. In order to prove the previous results, it is also shown that, if X _n converges in distribution to a separable limit, then X _{n k} converges stably for some subsequence (n _k ).